1. Field
The present invention relates to simulation usable for in-depth investigation of thermodynamic processes.
2. Description of the Related Art
Generally, according to the time-scale and to the physical scale which is to be simulated, different simulation techniques are suitable. FIG. 1 is a diagrammatic illustration of different kinds of modelling used as the time for modelling and size of the model increase. At the lower end, quantum mechanics takes more of a scientific rather than a practical approach to model in the area of Angstrom units and picoseconds. Generally this area of modelling deals with individual electrons and molecular mechanics deals with atoms. Continuing up the scale, coarse-grained modelling refers to models in which a few atoms which are close in terms of their properties are considered together as one “particle” (or group of atoms determined by the simulation parameters). Approximately the same approach can be taken with meso-scale modelling, in which maybe hundreds of atoms can be clustered to form one particle. Coarse-grained modelling is appropriate from sizes of a few Angstrom units to sizes of a few micrometers. Meso-scale modelling on the other hand can be interpreted as modelling on the scale of tens of nanometers to millimeters. The reader will appreciate the overlap between these two approximate scales. Finally, finite element analysis works on a continuum basis rather than with particles and is a more practical way of investigating properties of larger systems.
The present application is particularly, although not exclusively, concerned with coarse-grained and meso-scale modelling. These scales can be seen as the transitional regions between macroscopic and microscopic regimes. In such areas, atomistic methods such as molecular dynamics can be too expensive, whereas continuum solvers such as finite element analysis neglect the microstructure, which could lead to inaccurate results.
There are many phenomena which occur at meso-scales and merit careful study using simulation. Fluid mixture properties, such as emulsions, surfactants and phase separation in complex fluids can be investigated at meso-scale. Colloid suspensions with their aggregation clustering and dispersion are another area of interest. Also, the characteristics of a polymeric solution, such as melting characteristics and the behaviour of a dense solution are well suited to meso-scale modelling. These are just a few of the areas of application.
FIG. 2 illustrates different meso-scale modelling techniques currently proposed for meso-scale modelling. Of these, the most common is probably Dissipative Particle Dynamics.
Dissipative particle dynamics (DPD) has become a powerful and popular method to perform meso-scale simulations. DPD represents an intermediate position between all-atom molecular dynamics (MD) and Navier-Stokes equations. As its name suggests, DPD is particle based. Its computational cost scales linearly with the number of particles if the DPD algorithm is properly implemented, and hence very large systems can be simulated. The method can be used in complex-geometry domains. On a mathematical level, DPD generally predicts the behaviour of systems consisting of particles which are interacting through a combination of conservative, dissipative and fluctuation forces. Newton's laws are thus observed. Moreover, DPD can give an accurate prediction of hydrodynamic behaviour.
Despite its advantages, DPD has certain practical problems. Commonly used integration schemes in DPD lead to distinct deviations from the true equilibrium behaviour, including deviations from the temperature predicted by the fluctuation-dissipation theorem. None of the existing numerical implementations of DPD can reproduce correctly the simulation temperature under the full DPD dynamics. Thus, increases in the time-step used lead to a higher temperature and changes in all the thermodynamic properties dependent on temperature. Since the fluctuation-dissipation terms in DPD can be comparable to the conservative contributions, the non-preservation of thermodynamic equilibrium properties poses a serious obstacle for practical simulations.
A similar problem arises in classical molecular simulations when performing simulations under constant temperature.
Specifying the temperature in molecular dynamics (MD) simulations for example, involves a thermostat that represents the coupling of the molecular degrees of freedom to a “heatbath”. Thermostats can be categorized as either local or global. The simplest local thermostat is provided by Andersen's thermostat (Andersen, 1980), while the most common global thermostat is the Nosé-Hoover thermostat (Hoover, 1985).
From a physical point of view the local approach seems more realistic since it avoids a global coupling of all molecular degrees of freedom through extended “heatbath” variables. Rigorous constant-temperature sampling methods have been devised in the context of Monte Carlo methods, and a thermodynamically consistent implementation (i.e. free of numerical time-stepping artifacts) of Andersen's thermostat is provided by the hybrid Monte Carlo (HMC) method (Duane et al., 1987) and the generalized hybrid Monte Carlo (GHMC) method (Kennedy & Pendleton, 2001).
These methods are based on a hybrid of two long-established molecular simulation methods, molecular dynamics (MD) and Monte Carlo (MC). In MD, particles interact deterministically over a time period under known laws of physics whereas in MC conformations are accepted (or rejected) with a probability governed by a so-called Metropolis test involving positions and momenta.
The computational efficiency of HMC has been improved through the work of Izaguirre and co-workers (Izaguirre & Hampton, 2004; Sweet et al., 2006). Similar improvements have been achieved for the GHMC method by Akhmatskaya & Reich (2006, 2008), which have led to the generalized shadow hybrid Monte Carlo (GSHMC) method (Akhmatskaya & Reich, 2008).
In GSHMC the acceptance rate of the dynamics part of the GHMC is improved through the use of modified energies in the Metropolis test. The GSHMC method allows for efficient sampling of phase space for large molecular systems and can be used as a powerful simulation tool in a wide range of applications. It outperforms other popular simulation techniques such as classical MD and the standard hybrid MC in terms of sampling efficiency.
Even though these molecular simulation methods provide thermodynamically consistent implementations of constant-temperature molecular dynamics, they are not suitable for meso-scale simulations since the fluctuation-dissipation contributions are not applied in a dynamically consistent manner. The reason for this is that the momentum refreshment step of GHMC/GSHMC does not respect the Galilean invariance (Newton's third law) of the underlying force fields. Galilean invariance is a principle of relativity which states that the fundamental laws of physics are the same in all inertial frames. Galilean invariance is one of the key requirements for simulation methods adopted in meso-scale modelling, because the collective motion of the particles at this scale is more important, so that it is the co-operative nature of the simulated system which requires modelling.
Most local thermostats do not respect the Galilean invariance of the molecular force field, which implies conservation of total and angular momentum. This limitation has been overcome by the Lowe-Peters-Andersen thermostat (Lowe, 1999; Peters, 2004). It has also been found (Koopman & Lowe, 2006) that the Lowe-Peters-Andersen thermostat reduces the artificially induced viscosity compared to the Andersen thermostat at equal collision rates, which implies faster diffusion of particles in phase space. However, the Lowe-Peters-Andersen method cannot reproduce correctly thermodynamic quantities independently of time step in MD under DPD.
It is desirable to overcome the disadvantages of the prior art, particularly in the coarse-grain and meso-scale simulation areas.